We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.
Edge Universality for Orthogonal Ensembles of Random Matrices
Universality at the edge of the spectrum for unitary, orthogonal and symplectic ensembles of random matrices
Universal sum and product rules for random matrices
